Recalling from Section 4, the propagation of light in a crystal can be described completely by the impermeability tensor $\mu_{i,j}$, or $\mu = \epsilon_0\epsilon^{-1}$. To find the modes where light propagates through undisturbed, the index ellipsoid can be used to most easily identify the directions and the velocity that the components of the incident wave will propagate. The index ellipsoid in the principle coordinate system is given as:

\begin{equation} \frac{x^2}{n_x^2} + \frac{y^2}{n_y^2} + \frac{z^2}{n_z^2} = 1 \end{equation}

where x, y, and z are the directions of the crystal where $D$ and $E$ are parallel. $1/n_x^2$, $1/n_y^2$, and $1/n_z^2$ are the principle values of the impermeability tensor $\mu_{i,j}$.

Here is the interesting part! The electro-optic effect can be explained using solid state physics or quantum mechanics. The basic idea is that crystals are made up of periodically spaced atoms which are bonded together through some form of giving up or sharing electrons. One can then imagine that if light (an electromagnetic wave, of course) is used to pump a laser and induces lasing via the excitation of atoms in a medium, then if we have an electric field also exciting atoms in a material, then something interesting happens. In this case, the crystals chosen to demonstrate the electro-optic effect don’t support lasing easily, but in the presence of an electric field their bonding lengths and strengths are changed, meaning that the dielectric constant and thus the impermeability tensor are also changed. The net change to the impermeability tensor can be described by the equation:

\begin{equation} \mu_{i,j}(E) – \mu_{i,j}(0) = \Delta\mu_{i,j} = r_{i,j}E_k + s_{i,j,k,l}E_kE_l \end{equation}

where the choice of “$r_{i,j}$” and “$s_{i,j,k,l}$” is to be consistent with Yariv and Yeh. The term “$r_{i,j}$” is known as the linear (Pockels) electro-optic coefficient and “$s_{i,j,k,l}$” is known as the quadratic (Kerr) electro-optic coefficient. The above equation represents an expansion, but terms higher than the quadratic are not included as they would pose a negligible effect.

It is helpful to note which materials can exhibit the linear vs quadratic electro-optic effect to further understand this phenomenon. The linear electro-optic effect occurs in materials without inversion symmetry, while the quadratic electro-optic effect can occur in materials with any symmetry. This is thus why only the quadratic electro-optic effect is considered in centro-symmetric materials. We can prove that the inversion symmetry causes the linear electro-optic effect to disappear by considering the transformation of a point at $r$ about the inversion center to the position -$r$. Now, for the linear electro-optic tensor, a spatial transformation would also affect the tensor such that:

\begin{equation} Ir_{i,j,k} = r’_{i,j,k} = -r_{i,j,k} \label{inverse}\end{equation}

But then, it is also the property of tensors that any tensor property must be invariant under the operation of an inversion so that we have:

\begin{equation} r’_{i,j,k} = r_{i,j,k} {inverse_sym}\end{equation}

Since Equations \eqref{inverse} and \eqref{inverse_sym} can only be simultaneously satisfied if $r_{i,j,k} = 0$, then the linear electro-optic effect must vanish in centrosymmetric materials. In materials lacking the inversion symmetry, Equations \eqref{inverse} and \eqref{inverse_sym} would not be applicable, so the problem with inversion symmetry would not apply.

The quadratic electro-optic effect is normally neglected in materials that exhibit the linear electro-optic effect as the quadratic electro-optic coefficients will be much smaller. Due to the fact that the quadratic electro-optic effect can occur in materials with any symmetry, it is not surprising to note that the quadratic electro-optic effect was demonstrated first by John Kerr in 1875. His experiments were done with optically isotropic media such as liquids and glasses. In the presence of an electric field, these materials behave like a uniaxial crystal whose principle axes depend on the applied field.

Due to the fact that $\mu_{i,j}$ is a symmetric tensor, the “i” and “j” may be permuted. Similarly for “k” and “l”, the quadratic electro-optic coefficient is given as:

\begin{equation} s_{i,j,k,l} = \frac{1}{2}(\frac{\partial^2\mu_{i,j}}{\partial{E_k}\partial{E_l}})_{E=0} \end{equation}

Since the order of the partial differentiation doesn’t matter, the indices “k” and “l” can be permuted.